Supported by the Natural Science Foundation of China (11575195, 11875053), Y.L. also acknowledges the support from Jiangxi Young Scientists (JingGang Star) Program and 555 Talent Project of Jiangxi Province

^{3}and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

We investigate the bipartite entanglement for the boundary states in a simple type of spin networks with dangling edges, in which the two complementary parts are linked by two or more edges. Firstly, the spin entanglement is considered in the absence of the intertwiner entanglement. By virtue of numerical simulations, we find that the entanglement entropy usually depends on the group elements. More importantly, when the intertwiner entanglement is taken into account, we find that it is in general impossible to separate the total entanglement entropy into the contribution from spins on edges and the contribution from intertwiners at vertices. These situations are in contrast to the case when the two vertices are linked by a single edge.

Article funded by SCOAP^{3}

Entanglement is the prominent phenomenon in quantum physics. Recently, it has been discovered that it also plays a key role in understanding the emergence of spacetime in the framework of holographic gravity [

Above attempts of investigating the structure of spacetime by entanglement are background dependent. In particular, the RT formula is proposed in the large

The separation of spin entanglement and intertwiner entanglement in a network looks peculiar if one recalls the nonlinear nature of entanglement entropy. One may speculate if it is always possible to separate the entropy into these two contributions in a general spin network. This clarification would improve our understanding of the structure of entanglement in spin networks. Therefore, in this paper we further develop the results of [

(color online) Sketch of the spin boundary state described by Eq.(_{1} = _{2} = 1. The entanglement entropy as a function of _{1} = _{2} = 1 (lower left); and _{1} = 2, _{2} = 1(lower right).

In this section, we evaluate the bipartite entanglement entropy for a few simple spin networks in the absence of intertwiner entanglement. First, we consider the case when the two neighboring vertices are linked by two edges directly. In general, a spin network is a graph Γ composed of edges and vertices, which could be closed or non-closed. The spin network state for a non-closed graph Γ with dangling edges _{e}_{o}_{v}_{o}_{e}_{v}_{o}_{o}_{e}_{o}^{j}^{∂}. Thus, a spin network state can be written as the direct product of two parts, namely

For simplicity, we first consider a spin network with only two vertices _{A}_{B}_{p}_{q}_{n}

The Hilbert space of the boundary spin states is

For numerical simulation we consider a specific example as shown in Fig. ^{mn}^{j−m}^{m,−n} is the virtual two-valent intertwiner denoting the direction of the holonomy. _{i}

We now consider the entanglement entropy for this bipartite system. We choose _{1}} and _{2}}, so that the reduced density matrix is given by _{A}_{B}(|Ψ_{J1;J2}〉〈 Ψ_{J1;J2}|). As a result, the entanglement entropy can be evaluated as
_{1} + 1)−ln(2_{2} + 1)|≤_{ρA}_{1} + 1) + ln(2_{2} + 1). In fact, in this special case, since there is only one dangling edge at each vertex, a stronger upper bound holds _{ρA}_{1} + 1), ln(2_{2} + 1)}. It is also interesting to note that the entanglement entropy vanishes for

(color online) Sketch of the boundary state described by Eq.(_{i}_{i}_{1} = _{3} = 2, _{2} = _{4} = 1 (lower right).

Next, we consider the case that two vertices are linked by more than one path, which means that some paths may connect them indirectly by passing through other vertices. This is, of course, a common case for general spin networks. As an example, we consider the spin network shown in Fig. _{1},_{4}} and _{2},_{3}} and _{A}_{B}(|Ψ_{J1 J4;J2 J3}〉〈 Ψ_{J1J4; J2 J3}|). Numerical results for a few specific spins are shown in Fig. _{1} and _{3}, respectively, we find that the entropy is bounded as |ln(2_{1} + 1)–ln(2_{3}+1) |≤_{ρA}(_{1} + 1)+ln(2_{3} + 1).

(color online) (1) Two neighboring vertices

(color online) (1) Two neighboring vertices _{1} and _{2}, respectively. (2) The extension of the orthogonal relation does not hold.

In this section, we take the intertwiner entanglement into account. In ref. [^{k2}〉 represents the _{2}-th component of the intertwiner state, and _{i}

This orthogonal relation can be represented as a diagram, as shown in Fig.

However, when two vertices are linked by two or more edges, we find that this situation does not hold any more. In general, the bipartite entanglement entropy can not be separated into a spin part and an intertwiner part. For explicitness, we consider two vertices _{1} and _{2}, respectively, as shown in Fig.

The above orthogonal relation is not a necessary condition for separating the intertwiner indices and spin indices. However, we remark that, in a general case, they can not be separated if two vertices are linked by more than one path. To support this statement, we evaluate the total entanglement entropy and the intertwiner entanglement entropy numerically for a few specific spin networks. An example is shown in Fig. _{1}, _{2} are possible spins on virtual edges inside intertwiner

The entanglement entropy _{I}

_{k1k2} |
||||||||
---|---|---|---|---|---|---|---|---|

0 | ln 4 | ln 4 | meaningless | 0 | ||||

_{I} |
0 | ln2 | 0 | 0 | ln2 | |||

If we take all spins on the dangling edges to be _{00} and _{11} are two components in the intertwiner space. It should be noted that the other two components _{01} and _{10} do not appear in the above equation simply because the contraction of the corresponding

Sketch of the boundary spin state with intertwiner entanglement described by Eq. (

The reduced density matrix for bipartition is given as _{M1M2} = Tr_{1}_{2}(|Ψ〉〈Ψ|). It is straightforward to obtain the
entanglement entropy, which is
_{k1k2},
_{I}

In this paper, we have investigated the bipartite entanglement for the boundary spin states in spin networks with dangling edges. In particular, we have constructed a simple type of spin network in which two complementary parts are linked by two paths, either in a direct or indirect manner. The numerical evaluation of entanglement entropy leads to the following two main results. Firstly, in the absence of the intertwiner entanglement, the entanglement entropy for the boundary state depends on the group elements of the holonomy, which can not be simply determined by the spins _{1} and _{2} on the edges connecting the complementary parts. Nevertheless, we have proposed a bound for the entanglement entropy, which is |ln(2_{1} + 1)–ln(2_{2} + 1)|≤_{1} + 1)+ln(2_{2} + 1). It would be very important to prove or test this bound in a general case. Secondly, when the intertwiner entanglement is taken into account, the total entanglement can not be written, in general, as the sum of intertwiner entanglement and spin entanglement, but as a mixture of these two contributions.

Although we have only considered the simple case with two paths connecting two vertices, we believe that the above statements could be applicable to more complicated cases in which two vertices are linked by more than two edges directly, or by indirect paths.

Finally, based on our current work it is quite intriguing to further explore the relationship between quantum entanglement and quantum geometry, described by spin network states in loop quantum gravity. Our investigation is in progress and will be published in the near future [

We are very grateful to Yuxuan Liu and Zhuoyu Xian for helpful discussions and suggestions.

Diagrammatic sketch of the processes in Eqs. (

Diagrammatic sketch of the processes in Eqs. (

In this Appendix, we demonstrate the absence of the orthogonal relation for intertwiners when two vertices are linked by two edges, namely the inequality in Eq. (

For convenience, we define the operator

A diagrammatic sketch of Eqs. (

On the one hand, by virtue of Eq. (

Next, we prove that the operators _{1} ⊗ _{2} ⊗ · · · _{H}_{p}, we have

If we contract both Eq. (_{1}· · · _{p}_{1}_{2}| ^{k2}〉, we get
_{2}, _{1},· · · _{p}_{1}, _{2}, such that _{1} + _{2})(_{1} + _{2} + 1) and |_{1}−_{2}|(|_{1} − _{2}| + 1). Therefore, our starting assumption is not true and the orthogonal relation as shown in Eq. (